Finite Group Databasebeta

← #368590

Subgroups — #368590

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1989C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×C4×C4⋊C8C211yesmaximal, characteristic
4#14709C211yesmaximal, characteristic
5C4×#389C211yesmaximal, characteristic
6#11316C211yesmaximal, characteristic
7#11315C211yesmaximal, characteristic
8#11293C211yesmaximal, characteristic
9#11292C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic
11#10880C211yesmaximal, characteristic
12#10850C211yesmaximal, characteristic
13#10850C211yesmaximal, characteristic
14#10820C211yesmaximal, characteristic
15#10820C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#1989—24no—
29#1989C418yescomplemented
30C4×C4⋊C8—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49#1436—21no—
50#1436—21no—
51#1436—21no—
52#1436—21no—
53#1436—21no—
54#1436—21no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C4×#95—21no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#125—22no—
67C2×#125—22no—
68C2×#125—22no—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125C2×C414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8C2×C414yes—
98C2×C4⋊C8D414yes—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—41no—
104C2×C4×C8—41no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389Q814yes—
109#382—22no—
110#382—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—44no—
115C4×C4⋊C4C2×C414yes—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C43—22no—
135C43—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—41no—
145#340—41no—
146#340—41no—
147#340—41no—
148#340C2×C414yes—
149#340C2×C414yes—
150#340C2×C414yes—
151#340C2×C414yes—
152#336—22no—
153#336—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C22×C412yes—
194C22×C8C22×C412yes—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197#125—28no—
198#125—28no—
199#125—44no—
200#125—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C2×Q812yes—
258C2×C4⋊C4C2×Q812yes—
259C2×C4⋊C4#4112yes—
260C2×C4⋊C4#4112yes—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C22×C412yes—
290C2×C42C2×D412yes—
291C2×C42C2×Q812yes—
292C2×C42#4112yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297C4⋊C8—28no—
298C4⋊C8—28no—
299C4×C8—44no—
300C4×C8—44no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—42no—
319C24C22×D411yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4C22⋊Q812yes—
409C22×C4C22⋊Q812yes—
410C22×C4#12412yes—
411C22×C4#12412yes—
412C22×C4#13512yes—
413C22×C4#13512yes—
414C22×C4#13512yes—
415C22×C4#13512yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4#13712yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4H(5,𝔽2)11yescharacteristic
429C22×C42−1+411yescharacteristic
430C22×C42−1+411yescharacteristic
431C22×C42−1+411yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—48no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4C4×Q814yes—
496C42—28no—
497C42—28no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—44no—
503C42—44no—
504C42—48no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—84no—
510C42#11714yes—
511C42C4×D414yes—
512C42#12314yes—
513C42#12614yes—
514C23C2×C4×D411yescharacteristic
515C23C2×#12111yescharacteristic
516C23C2×C22⋊Q811yescharacteristic
517C23#52511yescharacteristic
518C23C2×#12411yescharacteristic
519C23#53611yescharacteristic
520C23#53711yescharacteristic
521C23C2×#13511yescharacteristic
522C23C2×#13511yescharacteristic
523C23C2×#13511yescharacteristic
524C23C2×#13511yescharacteristic
525C23C2×AGL(1,ℤ/8)11yescharacteristic
526C23C2×#13711yescharacteristic
527C23#57511yescharacteristic
528C23#57511yescharacteristic
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4#38814yes—
624C2×C4#38814yes—
625C2×C4#47814yes—
626C2×C4#47914yes—
627C2×C4C2×#11711yescharacteristic
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×C4×Q811yescharacteristic
630C2×C4C4×#4111yescharacteristic
631C2×C4#51911yescharacteristic
632C2×C4#52011yescharacteristic
633C2×C4C2×#12311yescharacteristic
634C2×C4C2×#12611yescharacteristic
635C2×C4#52911yescharacteristic
636C2×C4#53311yescharacteristic
637C2×C4#54111yescharacteristic
638C2×C4#54211yescharacteristic
639C2×C4#54611yescharacteristic
640C2×C4#54711yescharacteristic
641C2×C4#54811yescharacteristic
642C2×C4D4×Q811yescharacteristic
643C2×C4#55211yescharacteristic
644C2×C4#55211yescharacteristic
645C2×C4#55411yescharacteristic
646C2×C4#55611yescharacteristic
647C2×C4#55611yescharacteristic
648C2×C4#55611yescharacteristic
649C2×C4#55611yescharacteristic
650C2×C4#55611yescharacteristic
651C2×C4#55711yescharacteristic
652C2×C4#55711yescharacteristic
653C2×C4#55911yescharacteristic
654C2×C4#55911yescharacteristic
655C8—42no—
656C8—42no—
657C8—42no—
658C8—42no—
659C22C2×#38811yescharacteristic
660C22C4×C22⋊Q811yescharacteristic
661C22#232311yescharacteristic
662C22#241511yescharacteristic
663C22#241711yescharacteristic
664C22#256811yescharacteristic
665C22#259211yescharacteristic
666C22C4×#13511yescharacteristic
667C22C4×#13511yescharacteristic
668C22C4×AGL(1,ℤ/8)11yescharacteristic
669C22C4×#13711yescharacteristic
670C22#305011yescharacteristic
671C22#305011yescharacteristic
672C22#305511yescharacteristic
673C22#305611yescharacteristic
674C22C2×#47811yescharacteristic
675C22C2×#47911yescharacteristic
676C22#307611yescharacteristic
677C22#307611yescharacteristic
678C22#310211yescharacteristic
679C22#310211yescharacteristic
680C22#310511yescharacteristic
681C22#310811yescharacteristic
682C22#313111yescharacteristic
683C22#313111yescharacteristic
684C22#314111yescharacteristic
685C22#314211yescharacteristic
686C22#322811yescharacteristic
687C22#323111yescharacteristic
688C22#323211yescharacteristic
689C22#323211yescharacteristic
690C22#324711yescharacteristic
691C22#324811yescharacteristic
692C22#325011yescharacteristic
693C22#325011yescharacteristic
694C4—24no—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—161no—
707C4—161no—
708C4#235511yescharacteristic
709C4#235611yescharacteristic
710C4#236511yescharacteristic
711C4#236711yescharacteristic
712C4#244611yescharacteristic
713C4#244711yescharacteristic
714C4#251111yescharacteristic
715C4#251411yescharacteristic
716C2#1402111yescharacteristic
717C2#2108311yescharacteristic
718C2#2109311yescharacteristic
719C2C4×#47811yescharacteristic
720C2C4×#47911yescharacteristic
721C2#2211411yescharacteristic
722C2#2211511yescharacteristic
723C2#2470411yescharacteristic
724C2#2470511yescharacteristic
725C2#2483911yescharacteristic
726C2#2484011yescharacteristic
727C2#2922011yescharacteristic
728C2#2922111yescharacteristic
729C2#2935011yescharacteristic
730C2#2935111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
29. #1989C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#19892
C2×C4×C4⋊C82
#147092
C4×#3892
#113162
#113152
#112932
#112922
#108802
#108502
#108202